3.203 \(\int \frac{a+b \cosh ^{-1}(c+d x)}{(c e+d e x)^{3/2}} \, dx\)

Optimal. Leaf size=84 \[ \frac{4 b \sqrt{-c-d x+1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right ),-1\right )}{d e^{3/2} \sqrt{c+d x-1}}-\frac{2 \left (a+b \cosh ^{-1}(c+d x)\right )}{d e \sqrt{e (c+d x)}} \]

[Out]

(-2*(a + b*ArcCosh[c + d*x]))/(d*e*Sqrt[e*(c + d*x)]) + (4*b*Sqrt[1 - c - d*x]*EllipticF[ArcSin[Sqrt[e*(c + d*
x)]/Sqrt[e]], -1])/(d*e^(3/2)*Sqrt[-1 + c + d*x])

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Rubi [A]  time = 0.0839204, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {5866, 5662, 117, 116} \[ \frac{4 b \sqrt{-c-d x+1} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right )\right |-1\right )}{d e^{3/2} \sqrt{c+d x-1}}-\frac{2 \left (a+b \cosh ^{-1}(c+d x)\right )}{d e \sqrt{e (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[c + d*x])/(c*e + d*e*x)^(3/2),x]

[Out]

(-2*(a + b*ArcCosh[c + d*x]))/(d*e*Sqrt[e*(c + d*x)]) + (4*b*Sqrt[1 - c - d*x]*EllipticF[ArcSin[Sqrt[e*(c + d*
x)]/Sqrt[e]], -1])/(d*e^(3/2)*Sqrt[-1 + c + d*x])

Rule 5866

Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
]

Rule 5662

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcC
osh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 117

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[(Sqrt[1 + (d*x)/c]
*Sqrt[1 + (f*x)/e])/(Sqrt[c + d*x]*Sqrt[e + f*x]), Int[1/(Sqrt[b*x]*Sqrt[1 + (d*x)/c]*Sqrt[1 + (f*x)/e]), x],
x] /; FreeQ[{b, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 0])

Rule 116

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d), 2]*E
llipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-(b/d), 2])], (c*f)/(d*e)])/(b*Sqrt[e]), x] /; FreeQ[{b, c, d, e, f}, x]
&& GtQ[c, 0] && GtQ[e, 0] && (PosQ[-(b/d)] || NegQ[-(b/f)])

Rubi steps

\begin{align*} \int \frac{a+b \cosh ^{-1}(c+d x)}{(c e+d e x)^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}(x)}{(e x)^{3/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac{2 \left (a+b \cosh ^{-1}(c+d x)\right )}{d e \sqrt{e (c+d x)}}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x} \sqrt{e x} \sqrt{1+x}} \, dx,x,c+d x\right )}{d e}\\ &=-\frac{2 \left (a+b \cosh ^{-1}(c+d x)\right )}{d e \sqrt{e (c+d x)}}+\frac{\left (2 b \sqrt{1-c-d x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} \sqrt{e x} \sqrt{1+x}} \, dx,x,c+d x\right )}{d e \sqrt{-1+c+d x}}\\ &=-\frac{2 \left (a+b \cosh ^{-1}(c+d x)\right )}{d e \sqrt{e (c+d x)}}+\frac{4 b \sqrt{1-c-d x} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right )\right |-1\right )}{d e^{3/2} \sqrt{-1+c+d x}}\\ \end{align*}

Mathematica [C]  time = 0.155461, size = 92, normalized size = 1.1 \[ \frac{2 \left (\frac{2 b (c+d x) \sqrt{1-(c+d x)^2} \text{Hypergeometric2F1}\left (\frac{1}{4},\frac{1}{2},\frac{5}{4},(c+d x)^2\right )}{\sqrt{c+d x-1} \sqrt{c+d x+1}}-a-b \cosh ^{-1}(c+d x)\right )}{d e \sqrt{e (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCosh[c + d*x])/(c*e + d*e*x)^(3/2),x]

[Out]

(2*(-a - b*ArcCosh[c + d*x] + (2*b*(c + d*x)*Sqrt[1 - (c + d*x)^2]*Hypergeometric2F1[1/4, 1/2, 5/4, (c + d*x)^
2])/(Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])))/(d*e*Sqrt[e*(c + d*x)])

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Maple [A]  time = 0.016, size = 119, normalized size = 1.4 \begin{align*} 2\,{\frac{1}{de} \left ( -{\frac{a}{\sqrt{dex+ce}}}+b \left ( -{\frac{1}{\sqrt{dex+ce}}{\rm arccosh} \left ({\frac{dex+ce}{e}}\right )}+2\,{\frac{1}{e}{\it EllipticF} \left ( \sqrt{dex+ce}\sqrt{-{e}^{-1}},i \right ) \sqrt{-{\frac{dex+ce-e}{e}}}{\frac{1}{\sqrt{-{e}^{-1}}}}{\frac{1}{\sqrt{{\frac{dex+ce-e}{e}}}}}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(d*x+c))/(d*e*x+c*e)^(3/2),x)

[Out]

2/d/e*(-a/(d*e*x+c*e)^(1/2)+b*(-1/(d*e*x+c*e)^(1/2)*arccosh(1/e*(d*e*x+c*e))+2/e*EllipticF((d*e*x+c*e)^(1/2)*(
-1/e)^(1/2),I)*(-(d*e*x+c*e-e)/e)^(1/2)/(-1/e)^(1/2)/((d*e*x+c*e-e)/e)^(1/2)))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(d*x+c))/(d*e*x+c*e)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d e x + c e}{\left (b \operatorname{arcosh}\left (d x + c\right ) + a\right )}}{d^{2} e^{2} x^{2} + 2 \, c d e^{2} x + c^{2} e^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(d*x+c))/(d*e*x+c*e)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*e*x + c*e)*(b*arccosh(d*x + c) + a)/(d^2*e^2*x^2 + 2*c*d*e^2*x + c^2*e^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \operatorname{acosh}{\left (c + d x \right )}}{\left (e \left (c + d x\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acosh(d*x+c))/(d*e*x+c*e)**(3/2),x)

[Out]

Integral((a + b*acosh(c + d*x))/(e*(c + d*x))**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arcosh}\left (d x + c\right ) + a}{{\left (d e x + c e\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(d*x+c))/(d*e*x+c*e)^(3/2),x, algorithm="giac")

[Out]

integrate((b*arccosh(d*x + c) + a)/(d*e*x + c*e)^(3/2), x)